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SubjectsMathematics
समान समुच्चय और उपसमुच्चय
In this Class 10 Mathematics topic from the chapter Sets, students learn how to determine when two sets are equal by comparing their elements, regardless of the order in which those elements are written. They also study subsets, proper subsets, and the meaning of symbols such as ⊆ and ⊂. Clear examples help students test set relationships, identify all possible subsets of a set, and distinguish between equal, equivalent, and different sets.
TOPIC PRACTICE
Up to 6 questions from this page. Select your focus, then start.
Consider one particular element of A. Because X ⊆ Y, that element has exactly three permissible statuses: it may belong to neither X nor Y, it may belong to Y but not X, or it may belong to both X and Y. The status “in X but not in Y” is forbidden. Since there are 3 independent elements and 3 choices for each, the total number of ordered pairs is 3 × 3 × 3 = 3³ = 27. Thus option C is correct.
The assertion is true because every subset of \(A\) is automatically a subset of \(B\) when \(A\subset B\). Hence every element of \(\mathcal{P}(A)\) belongs to \(\mathcal{P}(B)\). The inclusion is proper, not merely equal, because \(B\subseteq B\), so \(B\in\mathcal{P}(B)\); however, \(B\notin\mathcal{P}(A)\), since \(B\) is not a subset of the smaller set \(A\). Thus both the assertion and reason are correct, and the reason explains the assertion.
The empty set ∅ is explicitly listed as an element of A, so ∅ ∈ A. Also, the empty set is a subset of every set because it has no element that could violate the subset condition; hence ∅ ⊆ A. Therefore, ∅ satisfies both requirements. Although 2 is an element of A, it is not a set and therefore is not a subset of A in this context.
A subset contains only elements that belong to A. Since 3 ∈ A, the set {3} is a subset of A. However, the listed elements of A are ∅, {∅}, and 3; the set {3} is not among them, so {3} ∉ A. The empty set, {∅}, and 3 are all elements of A, so they do not satisfy the phrase “not an element of A.”
The elements 1 and 3 are odd, while 2 and 4 are even. A subset has an even total when it contains either zero or two odd elements. Among the 16 subsets of A, exactly half have an even sum and half have an odd sum. Therefore, the number of subsets with even sum is 16 ÷ 2 = 8. The empty subset is included because its sum is 0, which is even.
For each element of A, the condition X ⊆ Y allows exactly three possibilities: the element is in neither X nor Y, it is in Y but not X, or it is in both X and Y. It cannot be in X alone because every element of X must also belong to Y. Since A has three independent elements, the total number of ordered pairs is 3 × 3 × 3 = 3³ = 27.
QUIZ COMPLETE
